Tires make a little difference. If you're PSI is lower you grab more. Side tread can help.
Experience with approach (ie, turning larger to hit the actual slippery part straighter)
Not holding the rear brake while turning (biggie)
A trick MTB use is to stand in your saddle and push the bike down to the inner part of the turn, but stay upright over your rear tire to help it stick better.
If you instead lean with the bike & turn you will be creating an outward force for the wheel to slide rather than a downward force the stick the tire to the surface.

From this point on, its all about going back there, studying the surface, studying your approach and figuring what you could do better after a field observation.
THEN its the long slow road toward rebuilding up your confidence by approaching at slower speeds, working on technique. What you gain in grabby tires, you lose in rolling resistance in keeping up.
And even the stickiest tires will slide in poor conditions or poor riding skills.

My advice is go with 28-38, keep PSI lower than average (ie 90 psi for 28") and work on putting weight over the wheel if you see a dangerous situation. Mind you, this isn't a roadies technique, just a Mountain Biker's technique applied to a dangerous situation coming upon you!

Wish your injury a speedy recovery. And a nice scar to talk about over beers!